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Quantum Field Theory

The SO(10) Group and Unification

The 16 fermions in the standard model are summarized in Table 06, where the superscript denotes electric charge, and the subscript denotes color charge. All the quarks in the SU(3) gauge group participate in strong interaction, all left-handed fermions undertake weak interaction, and those fermions carrying electric charge are eligible for electromagnetic interaction.

Gauge-Group /
SU(3) SU(2) U(1)
Left-handed ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3 0, e-1    
Right-handed ur2/3, uw2/3, ub2/3, dr-1/3, dw-1/3, db-1/3   e-1 0

Table 06 Fermions in Standard Model

An alternate arrangement to unify all these fermions into the 10 dimensional SO(10) group with the introduction of two color charges for weak interaction is proposed :
SO10 Group
  • There would be 5 generalized color charges including the red (r), green (g), blue (b) in strong interaction, and the yellow (y) and purple (p) in weak interaction.

  • Figure 05b lists all the fermions of the standard model in the first column. The generalized color charges are in the middle with the left-handed specie's denoted by solid circles, while the right-handed one's are represented by the anti-particles of the left-handed fermions, in hollow circles and carrying a "-" sign in front.

  • The color charges in strong interaction and those charges for the weak interaction are now lumped together to form the generalized hypercharge Y = - ( r + g + b ) / 3 + ( y + p ) / 2. This is not exactly the electric charge, for example, the left-handed neutrino and electron carries 0 and -1 electric charge respectively yielding the average value -1/2 for Y.
  • Figure 05b SO(10) Group
    [view large image]

  • The right-handed neutrino - is the odd-man/woman out in the standard model. It is there to provide a possible mechanism for the mass of the neutrinos. It does not involve in any of the interactions - that's why it has not been observed (smelling like dark matter ?).

  • The SO(10) group breaks into many schemes. One of them is the SU(3)SU(2)U(1) corresponding to the charges of (R, W, B), (G, P), and Y respectively. It is this particular grouping that fit all the elementary fermions snugly together without much room to maneuver. The breaking down to various subgroups occurred long time ago soon after the Big Bang.

  • In addition, the SO(10) has a 16-dimensional spinor representation, which fits exactly into the 16 fermions in Figure 05b.

  • To complete the list of all particles in the standard model, table 07 below displays all the force mediating bosons, where the single quote (') denotes anti-charge similar to the + charge of the positron in the electromagnetic interaction:

    Charge Gauge Field(s) Gauge Particle(s)
    r grr', grw', grb', gluon fields Gluons from red color charge
    w gww', gwr', gwb', gluon fields Gluons from white color charge
    b gbb', gbw', gbr', gluon fields Gluons from blue color charge
    g ggg', ggp', vector meson fields W vector mesons
    p gpp', gpg', vector meson fields Z0 vector meson
    e A, electromagnetic field Photon

    Table 07 Gauge Bosons

    There is actually only 8 independent bosons for SU(3) and 3 independent bosons for SU(2) since the trace (trace of a matrix A: TrA = ann) of their matrix representations is equal to zero - a condition to reduce the number of independent bosons by one. This is a requirement for the unitarity of the gauge group.

    Since the force mediating gluon itself carries color charges, the corresponding Feynman diagram is interpreted in slightly different way than those in QED. Figure 05e shows two quarks with blue and red colors moving toward each other initially. They interact by gluon exchange such
    Strong Interaction that the blue quark emits a blue-antired gluon thereby transforming itself into a red quark, while the red quark absorbs this gluon to become a blue quark. In other word, the gluon manages the creation and annihilation of the red-antired color charge pair, while fetches and donates the blue color charge from one quark to another resulting in the exchange of color charge on the quarks during the course of the interaction.

    Figure 05e Strong Interaction

    Since all the color charges are SO(10) group members that can be turned into each others, it is expected that at high probing energy, which
    Unification by SUSY enables penetration through the shielding by the virtual particles, the coupling strength of the various kinds of charges would merge into one unifying point. As shown by the diagram on the left of Figure 05d, it almost works, but not quite (the experimental errors is indicated by the width of the lines). The expectation comes true by introducing supersymmetry (SUSY), it even brings the outcast gravity to close proximity of the merging point (see diagram on the right of Figure 05d). This additional symmetry requires a new partner for all the particles.

    Figure 05d Unification by SUSY [view large image]

    They have the same charges (of all kinds) as their known partner, but different masses (heavier) and spins (integer 1/2 integer). Supersymmetry in effect doubles the number of particles in the standard model.

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